Anderson–Darling test

The Anderson - Darling test or Anderson -Darling goodness of fit test is a statistical test, which can be determined whether the frequency distribution of data from a sample of a given hypothetical probability distribution deviates. The best known and most common application of this adaptation is the use of tests as a normality test for the examination of a sample of normal distribution. It is named after the American mathematicians Theodore Wilbur Anderson and Donald Allan Darling, who first described it in 1952. More detailed studies of this test are by Michael A. Stephens.

Test Description

The Anderson -Darling test is based on a transformation sorted by size values ​​in the sample in a uniform distribution based on the distribution function of the given hypothetical probability distribution. As a test statistic functions of the distance of the transformed sample data on the distribution function of the uniform distribution. For a stronger weighting of the edge areas as well as for use in unknown expected values ​​and variances of various corrections are available. By Michael A. Stephens a method for direct estimation of the p- value is described in the test magnitude beyond. This is, with the distribution function of the given hypothetical probability distribution, calculated according to the formulas

With

The null hypothesis of the test is the assumption that the frequency distribution of the data in the sample of the given hypothetical probability distribution corresponds. A p-value less than 0.05 as a result of the Anderson -Darling test is thus to be interpreted as a significant deviation from the predetermined distribution. In contrast, represents a p-value greater than 0.05 but not necessarily, that the frequency distribution of data of the predetermined distribution corresponds to.

The Anderson -Darling test can be used from a sample size of n ≥ 8. Its most common application is the use as a normality test to compare the distribution of a sample with the normal distribution. The decision about whether the values ​​of a sample be normally distributed, is essential for the choice of the statistical tests for further analysis. While certain procedures such as the t - test and analysis of variance assume normally distributed samples could make deviations from the normal distribution, non-parametric tests such as the Mann-Whitney U test, the Wilcoxon signed - rank test, the Kruskal -Wallis test Friedman test or use as an alternative.

Alternative methods

An alternative to the Anderson - Darling test for general use as a goodness of fit test is the Kolmogorov -Smirnov test, which is also a sample of a hypothetical probability distribution can be compared. Compared to this the Anderson -Darling test considers certain critical values ​​, thereby having a higher sensitivity than the Kolmogorov -Smirnov test. However, these critical values ​​depend on the specific probability distribution, corresponding tables are currently available for the normal distribution, the lognormal distribution, the exponential distribution, the Weibull distribution, the type I extreme value distribution and the logistic distribution. The Kolmogorov -Smirnov test has the advantage that a comparison of the distribution of two samples is possible with it compared to the Anderson - Darling test. The same applies to the Cramér - von Mises test.

The Anderson - Darling test is one of the strongest statistical test methods for specific use as a normality test. Alternatives for this application are comparable in terms of statistical power in most cases, Shapiro - Wilk test, the Jarque - Bera test and also the Kolmogorov -Smirnov test as a test for normal distribution has, however, only a small test strength and compared to other normality tests is not recommended. Also, the Lilliefors test, in which it is a special adaptation of the Kolmogorov -Smirnov test to test for normal distribution, the Anderson - Darling test is inferior in strength test.